Lagrangian point

"Lagrange Point" redirects here. For the video game, see Lagrange Point (video game).
This article is about three-body libration points. For two-body libration points, see Geostationary orbit#Earth orbital libration points.

In celestial mechanics, the Lagrangian points (/ləˈɡrɑːndʒiən/; also Lagrange points, L-points, or libration points) are positions in an orbital configuration of two large bodies where a small object affected only by gravity can maintain a stable position relative to the two large bodies. The Lagrange points mark positions where the combined gravitational pull of the two large masses provides precisely the centripetal force required to orbit with them. There are five such points, labeled L1 to L5, all in the orbital plane of the two large bodies. The first three are on the line connecting the two large bodies and the last two, L4 and L5, form an equilateral triangle with the two large bodies. Objects can orbit these points as well.

Several planets have minor planets near their L4 and L5 points (trojans) with respect to the Sun, with Jupiter in particular having more than a million of these. Artificial satellites have been placed at L1 and L2 with respect to the Sun and Earth, and Earth and the Moon for various purposes, and the Lagrangian points have been proposed for a variety of future uses in space exploration.

Lagrange points in the Sun–Earth system (not to scale)

History

The three collinear Lagrange points (L1, L2, L3) were discovered by Leonhard Euler a few years before Lagrange discovered the remaining two.[1][2]

In 1772, Joseph-Louis Lagrange published an "Essay on the three-body problem". In the first chapter he considered the general three-body problem. From that, in the second chapter, he demonstrated two special constant-pattern solutions, the collinear and the equilateral, for any three masses, with circular orbits.[3]

Lagrange points

The five Lagrangian points are labeled and defined as follows:

The L1 point lies on the line defined by the two large masses M1 and M2, and between them. It is the most intuitively understood of the Lagrangian points: the one where the gravitational attraction of M2 partially cancels M1's gravitational attraction.

Explanation: An object that orbits the Sun more closely than Earth would normally have a shorter orbital period than Earth, but that ignores the effect of Earth's own gravitational pull. If the object is directly between Earth and the Sun, then Earth's gravity counteracts some of the Sun's pull on the object, and therefore increases the orbital period of the object. The closer to Earth the object is, the greater this effect is. At the L1 point, the orbital period of the object becomes exactly equal to Earth's orbital period. L1 is about 1.5 million kilometers from Earth.[4]

The L2 point lies on the line through the two large masses, beyond the smaller of the two. Here, the gravitational forces of the two large masses balance the centrifugal effect on a body at L2.

Explanation: On the opposite side of Earth from the Sun, the orbital period of an object would normally be greater than that of Earth. The extra pull of Earth's gravity decreases the orbital period of the object, and at the L2 point that orbital period becomes equal to Earth's. Like L1, L2 is about 1.5 million kilometers from Earth.

The L3 point lies on the line defined by the two large masses, beyond the larger of the two.

Explanation: L3 in the Sun–Earth system exists on the opposite side of the Sun, a little outside Earth's orbit but slightly closer to the Sun than Earth is. (This apparent contradiction is because the Sun is also affected by Earth's gravity, and so orbits around the two bodies' barycenter, which is, however, well inside the body of the Sun.) At the L3 point, the combined pull of Earth and Sun again causes the object to orbit with the same period as Earth.
Gravitational accelerations at L4

The L4 and L5 points lie at the third corners of the two equilateral triangles in the plane of orbit whose common base is the line between the centers of the two masses, such that the point lies behind (L5) or ahead (L4) of the smaller mass with regard to its orbit around the larger mass.

The triangular points (L4 and L5) are stable equilibria, provided that the ratio of M1/M2 is greater than 24.96.[note 1][5] This is the case for the Sun–Earth system, the Sun–Jupiter system, and, by a smaller margin, the Earth–Moon system. When a body at these points is perturbed, it moves away from the point, but the factor opposite of that which is increased or decreased by the perturbation (either gravity or angular momentum-induced speed) will also increase or decrease, bending the object's path into a stable, kidney-bean-shaped orbit around the point (as seen in the corotating frame of reference).

In contrast to L4 and L5, where stable equilibrium exists, the points L1, L2, and L3 are positions of unstable equilibrium. Any object orbiting at one of L1-L3 will tend to fall out of orbit; it is therefore rare to find natural objects there, and spacecraft inhabiting these areas must employ station keeping in order to maintain their position.

Natural objects at Lagrangian points

It is common to find objects at or orbiting the L4 and L5 points of natural orbital systems. These are commonly called "trojans"; in the 20th century, asteroids discovered orbiting at the Sun–Jupiter L4 and L5 points were named after characters from Homer's Iliad. Asteroids at the L4 point, which leads Jupiter, are referred to as the "Greek camp", whereas those at the L5 point are referred to as the "Trojan camp".

Other examples of natural objects orbiting at Lagrange points:

Mathematical details

A contour plot of the effective potential due to gravity and the centrifugal force of a two-body system in a rotating frame of reference. The arrows indicate the gradients of the potential around the five Lagrange points—downhill toward them (red) or away from them (blue). Counterintuitively, the L4 and L5 points are the high points of the potential. At the points themselves these forces are balanced.
Visualisation of the relationship between the Lagrangian points (red) of a planet (blue) orbiting a star (yellow) anticlockwise, and the effective potential in the plane containing the orbit (grey rubber-sheet model with purple contours of equal potential).[10]
Click for animation.

Lagrangian points are the constant-pattern solutions of the restricted three-body problem. For example, given two massive bodies in orbits around their common barycenter, there are five positions in space where a third body, of comparatively negligible mass, could be placed so as to maintain its position relative to the two massive bodies. As seen in a rotating reference frame that matches the angular velocity of the two co-orbiting bodies, the gravitational fields of two massive bodies combined with the minor body's centrifugal force are in balance at the Lagrangian points, allowing the smaller third body to be relatively stationary with respect to the first two.[11]

L1

The location of L1 is the solution to the following equation, balancing gravitation and the centrifugal force:

\frac{M_1}{(R-r)^2}=\frac{M_2}{r^2}+\left(\frac{M_1}{M_1+M_2}R-r\right)\frac{M_1+M_2}{R^3}

where r is the distance of the L1 point from the smaller object, R is the distance between the two main objects, and M1 and M2 are the masses of the large and small object, respectively. (The quantity in parentheses on the right is the distance of L1 from the center of mass.) Solving this for r involves solving a quintic function, but if the mass of the smaller object (M2) is much smaller than the mass of the larger object (M1) then L1 and L2 are at approximately equal distances r from the smaller object, equal to the radius of the Hill sphere, given by:

r \approx R \sqrt[3]{\frac{M_2}{3 M_1}}

This distance can be described as being such that the orbital period, corresponding to a circular orbit with this distance as radius around M2 in the absence of M1, is that of M2 around M1, divided by \sqrt{3}\approx 1.73:

T_{s,M_2}(r) = \frac{T_{M_2,M_1}(R)}{\sqrt{3}}.

L2

The location of L2 is the solution to the following equation, balancing gravitation and inertia:

\frac{M_1}{(R+r)^2}+\frac{M_2}{r^2}=\left(\frac{M_1}{M_1+M_2}R+r\right)\frac{M_1+M_2}{R^3}

with parameters defined as for the L1 case. Again, if the mass of the smaller object (M2) is much smaller than the mass of the larger object (M1) then L2 is at approximately the radius of the Hill sphere, given by:

r \approx R \sqrt[3]{\frac{M_2}{3 M_1}}

L3

The location of L3 is the solution to the following equation, balancing gravitation and the centrifugal force:

\frac{M_1}{(R-r)^2}+\frac{M_2}{(2R-r)^2}=\left(\frac{M_2}{M_1+M_2}R+R-r\right)\frac{M_1+M_2}{R^3}

with parameters defined as for the L1 and L2 cases except that r now indicates how much closer L3 is to the more massive object than the smaller object. If the mass of the smaller object (M2) is much smaller than the mass of the larger object (M1) then:

r \approx R \frac{7M_2}{12 M_1}

L4 and L5

The reason these points are in balance is that, at L4 and L5, the distances to the two masses are equal. Accordingly, the gravitational forces from the two massive bodies are in the same ratio as the masses of the two bodies, and so the resultant force acts through the barycenter of the system; additionally, the geometry of the triangle ensures that the resultant acceleration is to the distance from the barycenter in the same ratio as for the two massive bodies. The barycenter being both the center of mass and center of rotation of the three-body system, this resultant force is exactly that required to keep the smaller body at the Lagrange point in orbital equilibrium with the other two larger bodies of system. (Indeed, the third body need not have negligible mass.) The general triangular configuration was discovered by Lagrange in work on the three-body problem.

Stability

Although the L1, L2, and L3 points are nominally unstable, it turns out that it is possible to find (unstable) periodic orbits around these points, at least in the restricted three-body problem. These periodic orbits, referred to as "halo" orbits, do not exist in a full n-body dynamical system such as the Solar System. However, quasi-periodic (i.e. bounded but not precisely repeating) orbits following Lissajous-curve trajectories do exist in the n-body system. These quasi-periodic Lissajous orbits are what most of Lagrangian-point missions to date have used. Although they are not perfectly stable, a relatively modest effort at station keeping can allow a spacecraft to stay in a desired Lissajous orbit for an extended period of time. It also turns out that, at least in the case of Sun–Earth-L1 missions, it is actually preferable to place the spacecraft in a large-amplitude (100,000–200,000 km or 62,000–124,000 mi) Lissajous orbit, instead of having it sit at the Lagrangian point, because this keeps the spacecraft off the direct line between Sun and Earth, thereby reducing the impact of solar interference on Earth–spacecraft communications. Similarly, a large-amplitude Lissajous orbit around L2 can keep a probe out of Earth's shadow and therefore ensures a better illumination of its solar panels.

Spaceflight applications

Earth–Moon L1 allows comparatively easy access to Lunar and Earth orbits with minimal change in velocity and has this as an advantage to position a half-way manned space station intended to help transport cargo and personnel to the Moon and back.

Earth–Moon L2 would be a good location for a communications satellite covering the Moon's far side and would be "an ideal location" for a propellant depot as part of the proposed depot-based space transportation architecture.[12]

The satellite ACE in an orbit around L1

Sun–Earth L1 is suited for making observations of the Sun–Earth system. Objects here are never shadowed by Earth or the Moon. The first mission of this type was the International Sun Earth Explorer 3 (ISEE-3) mission used as an interplanetary early warning storm monitor for solar disturbances.

Sun–Earth L2 is a good spot for space-based observatories. Because an object around L2 will maintain the same relative position with respect to the Sun and Earth, shielding and calibration are much simpler. It is, however, slightly beyond the reach of Earth's umbra,[13] so solar radiation is not completely blocked. From this point, the Sun, Earth and Moon are relatively closely positioned together in the sky, and hence leave a large field of view without interference – this is especially helpful for infrared astronomy.

Sun–Earth L3 was a popular place to put a "Counter-Earth" in pulp science fiction and comic books. Once space-based observation became possible via satellites[14] and probes, it was shown to hold no such object. The Sun–Earth L3 is unstable and could not contain an object, large or small, for very long. This is because the gravitational forces of the other planets are stronger than that of Earth (Venus, for example, comes within 0.3 AU of this L3 every 20 months).

A spacecraft orbiting near Sun–Earth L3 would be able to closely monitor the evolution of active sunspot regions before they rotate into a geoeffective position, so that a 7-day early warning could be issued by the NOAA Space Weather Prediction Center. Moreover, a satellite near Sun–Earth L3 would provide very important observations not only for Earth forecasts, but also for deep space support (Mars predictions and for manned mission to near-Earth asteroids). In 2010, spacecraft transfer trajectories to Sun–Earth L3 were studied and several designs were considered.[15]

Scientists at the B612 Foundation are planning to use Venus's L3 point to position their planned Sentinel telescope, which aims to look back towards Earth's orbit and compile a catalogue of near-Earth asteroids.[16]

Missions to Lagrangian points generally orbit the points rather than occupy them directly.

Another interesting and useful property of the collinear Lagrangian points and their associated Lissajous orbits is that they serve as "gateways" to control the chaotic trajectories of the Interplanetary Transport Network.

Spacecraft at Sun–Earth L1

International Sun Earth Explorer 3 (ISEE-3) began its mission at the Sun–Earth L1 before leaving to intercept a comet in 1982. The Sun–Earth L1 is also the point to which the Reboot ISEE-3 mission was attempting to return the craft as the first phase of a recovery mission (as of September 25, 2014 all efforts have failed and contact was lost).[17]

Solar and Heliospheric Observatory (SOHO) is stationed in a halo orbit at L1, and the Advanced Composition Explorer (ACE) in a Lissajous orbit, also at the L1 point. WIND is also at L1.

Deep Space Climate Observatory (DSCOVR), launched on 11 February 2015, will orbit L1 to study the solar wind and its effects on Earth.

Spacecraft at Sun–Earth L2

Spacecraft at the Sun–Earth L2 point are in a Lissajous orbit until decommissioned, when they are sent into a heliocentric graveyard orbit.

List of missions to Lagrangian points

Color key:

     – Unflown or planned mission      – Mission en route or in progress (including mission extensions)      – Mission at Lagrangian point completed successfully (or partially successfully)

Past and present missions

Mission Lagrangian point Agency Description
International Sun–Earth Explorer 3 (ISEE-3) Sun–Earth L1 NASA Launched in 1978, it was the first spacecraft to be put into orbit around a libration point, where it operated for four years in a halo orbit about the L1 Sun–Earth point. After the original mission ended, it was commanded to leave L1 in September 1982 in order to investigate comets and the Sun.[22] Now in a heliocentric orbit, an unsuccessful attempt to return to halo orbit was made in 2014 when it made a flyby of the Earth–Moon system.[23][24]
Advanced Composition Explorer (ACE) Sun–Earth L1 NASA Launched 1997. Has fuel to orbit near the L1 until 2024. As of 2013 operational.[25]
Deep Space Climate Observatory (DSCOVR) Sun–Earth L1 NASA Launched on 11 February 2015. Intended to take 110 days from launch to reach L1. [26]
Solar and Heliospheric Observatory (SOHO) Sun–Earth L1 ESA, NASA Orbiting near the L1 since 1996. As of 2013 operational.[27]
WIND Sun–Earth L1 NASA Arrived at L1 in 2004 with fuel for 60 yrs. As of 2013 operational.[28]
Wilkinson Microwave Anisotropy Probe (WMAP) Sun–Earth L2 NASA Arrived at L2 in 2001. Mission ended 2010,[29] then sent to solar orbit outside L2.[30]
Herschel Space Observatory Sun–Earth L2 ESA Arrived at L2 July 2009. Ceased operation on 29 April 2013 and will be moved to a heliocentric orbit.[31][32]
Planck Space Observatory Sun–Earth L2 ESA Arrived at L2 July 2009. Mission ended on 23 October 2013, Planck has been moved to a heliocentric parking orbit.[33]
Chang'e 2 Sun–Earth L2 CNSA Original mission ended, left L2 point for 4179 Toutatis at April 15, 2012.[34]
ARTEMIS mission extension of THEMIS Earth–Moon L1 and L2 NASA Mission consists of two spacecraft, which were the first spacecraft to reach Earth–Moon Lagrangian points. Both moved through Earth–Moon Lagrangian points, and are now in lunar orbit.[35][36]
Gaia Sun–Earth L2 ESA Launched on 19 December 2013.[37][38]

Future and proposed missions

Mission Lagrangian point Agency Description
"Lunar Far-Side Communication Satellites" Earth–Moon L2 NASA Proposed in 1968 for communications on the far side of the Moon during the Apollo program, mainly to enable an Apollo landing on the far side—neither the satellites nor the landing were ever realized.[39]
Space colonization and manufacturing Earth–Moon L4 or L5 — First proposed in 1974 by Gerard K. O'Neill[40] and subsequently advocated by the L5 Society
LISA Pathfinder (LPF) Sun–Earth L1 ESA, NASA As of 2012 launch is scheduled for July 2015.[41]
Solar-C Sun–Earth L1 JAXA Possible mission after 2010.
James Webb Space Telescope (JWST) Sun–Earth L2 NASA, ESA, CSA As of 2013 launch is planned for October 2018.[42]
Euclid Sun–Earth L2 ESA, NASA As of 2013 planned for launch in 2020.[43]
Wide Field Infrared Survey Telescope (WFIRST) Sun–Earth L2 NASA, USDOE As of 2013 in a 'pre-formulation' phase until at least early 2016, possible launch in the early 2020s.[44]
Space Infrared Telescope for Cosmology and Astrophysics (SPICA) Sun–Earth L2 JAXA, ESA, SRON As of 2015 awaiting approval from both Japanese and European side, launch proposed for 2025.[45]
Exploration Gateway Platform Earth–Moon L2[46] NASA Proposed in 2011.[47]
Advanced Telescope for High Energy Astrophysics (ATHENA) Sun–Earth L2 ESA Launch planned for 2028[48]

The Aditya mission for a solar observatory, is officially scheduled for launch in 2016–17 by ISRO. "It will be going to a point 1.5 million kilometers away from Earth, that is L1 Lagrangian point. From this point it will observe the Sun constantly and study the solar corona, the region around the sun's surface.[49]

See also

Notes

  1. ↑ Actually \tfrac{25+\sqrt{621}}{2} ≈ 24.9599357944

References

  1. ↑ Koon, W. S.; M. W. Lo; J. E. Marsden; S. D. Ross (2006). Dynamical Systems, the Three-Body Problem, and Space Mission Design. p. 9. (16MB)
  2. ↑ Leonhard Euler, De motu rectilineo trium corporum se mutuo attrahentium (1765)
  3. ↑ Lagrange, Joseph-Louis (1867–92). "Tome 6, Chapitre II: Essai sur le problème des trois corps". Oeuvres de Lagrange (in French). Gauthier-Villars. pp. 229–334.
  4. ↑ Cornish, Neil J. "The Lagrangian Points" (PDF). Department of Physics, Bozeman Campus, Montana State University, USA. Retrieved 29 July 2011.
  5. ↑ The Lagrange Points PDF, Neil J. Cornish with input from Jeremy Goodman
  6. ↑ Space.com: First Asteroid Companion of Earth Discovered at Last
  7. ↑ NASA—NASA's Wise Mission Finds First Trojan Asteroid Sharing Earth's Orbit
  8. ↑ "A Search for Natural or Artificial Objects Located at the Earth–Moon Libration Points" by Robert Freitas and Francisco Valdes, Icarus 42, 442-447 (1980)
  9. ↑ "List Of Neptune Trojans". Minor Planet Center. Archived from the original on 2011-08-23. Retrieved 2010-10-27.
  10. ↑ ZF Seidov, "The Roche Problem: Some Analytics", The Astrophysical Journal, 603:283-284, 2004 March 1
  11. ↑ "Lagrange Points" by Enrique Zeleny, Wolfram Demonstrations Project.
  12. ↑ Zegler, Frank; Bernard Kutter (2010-09-02). "Evolving to a Depot-Based Space Transportation Architecture" (PDF). AIAA SPACE 2010 Conference & Exposition. AIAA. p. 4. Retrieved 2011-01-25. L2 is in deep space far away from any planetary surface and hence the thermal, micrometeoroid, and atomic oxygen environments are vastly superior to those in LEO. Thermodynamic stasis and extended hardware life are far easier to obtain without these punishing conditions seen in LEO. L2 is not just a great gateway—it is a great place to store propellants. ... L2 is an ideal location to store propellants and cargos: it is close, high energy, and cold. More importantly, it allows the continuous onward movement of propellants from LEO depots, thus suppressing their size and effectively minimizing the near-Earth boiloff penalties.
  13. ↑ Angular size of the Sun at 1 AU + 930000 miles: 31.6', angular size of Earth at 930000 miles: 29.3'
  14. ↑ STEREO mission description by NASA, http://www.nasa.gov/mission_pages/stereo/main/index.html#.UuG0NxDb-kk
  15. ↑ Tantardini, Marco; Fantino, Elena; Yuan Ren; Pierpaolo Pergola; Gerard Gómez; Josep J. Masdemont (2010). "Spacecraft trajectories to the L3 point of the Sun–Earth three-body problem". Celestial Mechanics and Dynamical Astronomy (Springer).
  16. ↑ The Sentinel Mission, B612 Foundation. Retrieved Feb 2014.
  17. ↑ "ISEE-3 is in Safe Mode". Space College. 25 September 2014. "The ground stations listening to ISEE-3 have not been able to obtain a signal since Tuesday the 16th"
  18. ↑ "Mission Complete! WMAP Fires Its Thrusters For The Last Time".
  19. ↑ Toobin, Adam (2013-06-19). "Herschel Space Telescope Shut Down For Good, ESA Announces". Huffington Post.
  20. ↑ Lakdawalla, Emily (14 June 2012). "Chang'E 2 has departed Earth's neighborhood for.....asteroid Toutatis!?". Retrieved 15 June 2012.
  21. ↑ Lakdawalla, Emily (15 June 2012). "Update on yesterday's post about Chang'E 2 going to Toutatis". Planetary Society. Retrieved 26 June 2012.
  22. ↑ "Solar System Exploration: ISEE-3/ICE". NASA. Retrieved 2010-09-28.
  23. ↑ Lakdawalla, Emily (October 3, 2008). "It's Alive!". The Planetary Science Weblog.
  24. ↑ Chang, Kenneth (August 8, 2014). "Rudderless Craft to Get Glimpse of Home Before Sinking Into Space's Depths". The New York Times.
  25. ↑ "ACE Mission". Caltech ACE Science Center. Retrieved 2013-03-18.
  26. ↑ http://www.spacex.com/news/2015/02/11/spacex-launches-dscovr-satellite-deep-space-orbit
  27. ↑ "SOHO's Orbit: An Uninterrupted View of the Sun". NASA. Retrieved 2010-09-28.
  28. ↑ "WIND Spacecraft". NASA. Retrieved 2010-09-28.
  29. ↑ "WMAP Facts". NASA. Retrieved 2013-03-18.
  30. ↑ http://map.gsfc.nasa.gov/news/events.html WMAP Ceases Communications
  31. ↑ "Herschel Factsheet". European Space Agency. 17 April 2009. Retrieved 2009-05-12.
  32. ↑ "Herschel space telescope finishes mission". BBC news. 29 April 2013.
  33. ↑ "Last command sent to ESA's Planck space telescope". European Space Agency. October 23, 2013. Retrieved October 23, 2013.
  34. ↑ "嫦娥二号再接新任务预计年底飞掠小行星". 现代快报. Retrieved 2012-06-27.
  35. ↑ Fox, Karen C. "First ARTEMIS Spacecraft Successfully Enters Lunar Orbit". The Sun-Earth Connection: Heliophysics. NASA.
  36. ↑ Hendrix, Susan. "Second ARTEMIS Spacecraft Successfully Enters Lunar Orbit". The Sun-Earth Connection: Heliophysics. NASA.
  37. ↑ "Worldwide launch schedule". Spaceflight Now. 27 November 2013.
  38. ↑ "ESA Gaia home". ESA. Retrieved 23 October 2013.
  39. ↑ P. E. Schmid (June 1968). "Lunar Far-Side Communication Satellites" (PDF). NASA. Retrieved 2008-07-16.
  40. ↑ O'Neill, Gerard K. (September 1974). "The Colonization of Space". Physics Today (American Institute of Physics) 27 (9): 32–40. Bibcode:1974PhT....27i..32O. doi:10.1063/1.3128863.
  41. ↑ "LISA Pathfinder factsheet". ESA. 11 June 2012. Retrieved 26 June 2012.
  42. ↑ "JWST factsheet". ESA. 2013-09-04. Retrieved 2013-09-07.
  43. ↑ "NASA Officially Joins ESA's 'Dark Universe' Mission". JPL/NASA. 24 January 2013. Retrieved 12 April 2013.
  44. ↑ Paul Hertz (2013-06-04), NASA Astrophysics presentation to American Astronomical Society (PDF), retrieved 2013-09-10
  45. ↑ Hiroshi Shibai (2014-12-31), SPICA (PDF), retrieved 2015-02-24
  46. ↑ NASA teams evaluating ISS-built Exploration Platform roadmap
  47. ↑ Bergin, Chris (December 2011). "Exploration Gateway Platform hosting Reusable Lunar Lander proposed". NASA Spaceflight.com. Retrieved 2011-12-05.
  48. ↑ "ESA Science & Technology: Athena to study the hot and energetic Universe". ESA. 27 June 2014. Retrieved 23 August 2014.
  49. ↑ http://www.business-standard.com/article/beyond-business/man-in-space-and-other-plans-114111401887_1.html

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